[Paper Review] On average population levels for models with directed diffusion in heterogeneous environments

André Rickes, Elena Braverman|arXiv (Cornell University)|Jan 7, 2026

Mathematical and Theoretical Epidemiology and Ecology Models0 citations

TL;DR

The paper analyzes how the total population at steady state responds to diffusion in a heterogeneous environment when the growth rate scales as r = K^λ, and also examines a three-parameter dispersal model with u/P diffusion, showing that the relationship between total population and carrying capacity depends on λ, P, and correlation between r and K/P.

ABSTRACT

In 2006 (J. Differential Equ.), Lou proved that, once the intrinsic growth rate $r$ in the logistic model is proportional to the spatially heterogeneous carrying capacity $K$ ($r=K^1$), the total population under the regular diffusion exceeds the total of the carrying capacity. He also conjectured that the dependency of the total population on the diffusion coefficient is unimodal, increasing to its maximum and then decreasing to the asymptote which is the total of the carrying capacity. DeAngelis et al (J. Math. Biol. 2016) argued that the prevalence of the population over the carrying capacity is only observed when the growth rate and the carrying capacity are positively correlated, at least for slow dispersal. Guo et al (J. Math. Biol. 2020) justified that, once $r$ is constant ($r=K^0$), the total population is less than the cumulative carrying capacity. Our paper fills up the gap for when $r=K^λ$ for any real $λ$, disproving an assumption that there is a critical $λ^{\ast} \in (0,1)$ at which the tendency of the prevalence of the carrying capacity over the total population size changes, demonstrating instead that the relationship is more complicated. In addition, we explore the dependency of the total population size on the diffusion coefficient when the third parameter of the dispersal strategy $P$ is involved: the diffusion term is $d Δ(u/P)$, not just $d Δu$, for any $λ$. We outline some differences from the random diffusion case, in particular, concerning the profile of the total population as a function of the diffusion coefficient.

Motivation & Objective

• Motivate understanding of how diffusion and spatial heterogeneity affect total population relative to carrying capacity.
• Extend prior results by analyzing r = K^λ for general λ and include a dispersal strategy P in the diffusion term.
• Investigate how total population M(d) depends on diffusion coefficient d across different λ and P configurations.
• Provide conditions under which total population exceeds or remains below carrying capacity and examine limits as d→0+ and d→∞.

Proposed method

• Study stationary solutions u_d of the generalized logistic equation with directed diffusion: d Δ(u/P) + r u (1 - u/K) = 0 with Neumann boundary conditions.
• Analyze limiting behavior of u_d as d → 0+ and d → ∞ to relate M(d) = ∫Ω u_d dx to ∫Ω K dx.
• Derive conditions under which M(d) > ∫Ω K dx for all d, using P = α K/r and non-constant r.
• Examine correlation between r and K/P to infer the sign of ∫Ω u_d dx − ∫Ω K dx for small d (and small perturbations).
• Extend to the case r = α (K/P)^λ and prove monotonicity of M_λ(+∞) with respect to λ.
• Provide examples illustrating scenarios where M(d) is monotone, unimodal, or more complex.

Figure 1: Population decline when $\int_{\Omega}\nabla(K/P)\cdot\nabla r\,dx=0$ , for $K(x)=(\cos(2\pi x)+2)^{2}$ , $P(x)=\sqrt{K(x)}$ , and $r(x)=\cos(\pi x)+2$ .

Experimental results

Research questions

• RQ1 For which dispersal strategies P does the total population M(d) exceed the carrying capacity ∫Ω K dx for all d>0?
• RQ2 How does the choice r = α (K/P)^λ influence the relation between M(d) and ∫Ω K dx, especially for small and large diffusion?
• RQ3 Under what conditions (on r, K, P) is M(d) unimodal in d or can it be non-unimodal?
• RQ4 What are the limiting behaviors of M(d) as d → 0+ and d → ∞ for the three-parameter dispersal model, and how do these relate to carrying capacity?
• RQ5 How does correlation between r and K/P affect low-d behavior of the total population?

Key findings

• If P = α K/r with non-constant r, then the total population exceeds carrying capacity for every d>0 (global inequality).
• When r is constant and P, K are non-constant and independent, the total population is always below carrying capacity for all d>0.
• For r = α K/r (i.e., P ∝ K/r), both d→0+ and d→∞ lead to M(d) approaching ∫Ω K dx, but M(d) is not necessarily unimodal in general.
• If r and K/P are positively correlated, the total population tends to exceed carrying capacity for small d; if negatively correlated, it tends to be below carrying capacity for small d.
• For r = α (K/P)^λ with λ > 0, the large-d limit M_λ(+∞) is strictly increasing in λ; this implies relative positioning of M_λ(0) and M_λ(+∞) depends on λ.
• The paper provides conditions and examples showing that M(d) can range from monotone to unimodal to more complex forms depending on P, K, r and λ.

Figure 2: Population growth when $\int_{\Omega}\nabla(K/P)\cdot\nabla r\,dx=0$ , for $K(x)=(2x^{3}-3x^{2}+3)e^{2x^{3}-3x^{2}}$ , $P(x)=e^{2x^{3}-3x^{2}}$ and $r(x)=\cos(2\pi x)+3$ .